Projective Varieties and Modular Forms Course Given at the Universit
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210 Martin Eichler UniversiUit Basel, Basel/Schweiz
Projective Varieties and Modular Forms Course Given at the University of Maryland, Spring 1970
Springer-Verlag Berlin· Heidelberg· New York 1971
AMS Subject Classifications (1970): l8G 10, 14M05, IOD 20
ISBN 3-54Q-55l9-3 Springer-Verlag Berlin . Heide1bt:rg . New York ISBN 0-38H)5519-3 Springer-Verlag New York· Heidelberg· Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin· Heidelberg 1971. Library of Congress Catalog Card Number 78-166998.
Offsetdruck: Julius Beitz, HemsbachlBergstr.
CONTENTS Introduction • . • • • .
1
Chapter I
4
Graded Modules
§l.
Some basic concepts
4
§2.
Free resolutions
9
3.
The functors
§4.
The functors
§5.
Modules over polynomial rings
§ 6.
The rank polynomial. .
§7.
Reduction of the number of variables
•
§ 8.
Structural properties.
• • 34
§9.
The theorem of Riemann-Roch and the duality theorem • • • 39
§
Chapter II
· • 13 18
continued
21 • 26
Graded Rings and Ideals •
• 3a
49
§10.
Introduction, divisors
§ll.
Differentials and the theorem of Riemann-Roch .
• 55
§l2.
Automorphic forms and projective varieties.
• 63
§13.
Quasiinvertible ideals
67
§14.
Intersection numbers
72
§15.
Regular local rings.
79
Chapter III
49
88
Applications to Modular Forms.
§l6.
Introduction
88
§l7.
Specializations of modular forms
91
§l8.
Principal ideals
§l9.
Hilbert modular forms in 2 variables . •
Appendix
..
Conjectures and Problems
I-
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• 97
100
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oil
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II
II
115
INTRODUCTION
W.L. BaiZy and A. BoreZ showed (Compactification of arithmetic quotients of bounded symmetric domains, Annals of Math. 84(1966), p. 442-528) that the graded rings of automorphic forms are finitely generated, provided that rather general conditions are satisfied.
Let
be a system of generators, then between the Y. some poly~
nomial equations
Fv(yo,···.ym)
=
0
will hold. and all these equations
are consequences of finitely many of them.
Let us assume for the sake
of simplicity (although it is wrong in general) that the Y same weights.
i
have the
Then the F (y.) are homogeneous polynomials, and their
v
~
zeros form a projective variety Gn in the projective space pm with the coordinates Yi' whose dimension is
n < m.
Projective varieties
are the most common examples of algebraic varieties although it can be shown that there exist (rather exceptional) algebraic varieties which are not projective.
The theorem of Baily and Borel opens the way to
an al
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